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7 - Logic in transition

Published online by Cambridge University Press:  05 March 2012

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Summary

I have long struggled against the admission of ranges of values thereby of classes; but I have found no other possibility to provide a logical foundation for arithmetic. This question is: how are we to conceive of logical objects? And I have found no answer other than this: we conceive of them as extensions of concepts or, more generally, as ranges of values of functions … what other way is there?

Frege to Russell, 28 July 1902

The first task in discussing the foundations of (pure) mathematics is to make precise the distinction between it and other sciences, a task which in Principia Mathematica is surprisingly neglected.

Ramsey, Undated manuscript (ASP)

Logicism and the foundational crisis

In 1900 Russell underwent the one event in his intellectual life that he was willing to characterize as a “revolution”: He met Peano and was struck by the capacity of Peano's work to shed light on the philosophical nature of mathematics. It was at this time that Russell conceived one of his most fruitful ideas, the logicist project.

Peano had identified a notational system, or a cluster of concepts, that seemed to have enormous expressive power. The hope was that it could be used to express all of mathematics, and Peano's school had been working for years at rewriting different fragments of mathematics in their peculiar notation. Russell suggested that its basic concepts might be reduced to purely “logical” notions, in an as yet undisclosed sense of that word, and that perhaps all the assumptions one needed were those of logic, whatever they might be.

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The Semantic Tradition from Kant to Carnap
To the Vienna Station
, pp. 113 - 140
Publisher: Cambridge University Press
Print publication year: 1991

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  • Logic in transition
  • J. Alberto Coffa
  • Book: The Semantic Tradition from Kant to Carnap
  • Online publication: 05 March 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139172240.009
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  • Logic in transition
  • J. Alberto Coffa
  • Book: The Semantic Tradition from Kant to Carnap
  • Online publication: 05 March 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139172240.009
Available formats
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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Logic in transition
  • J. Alberto Coffa
  • Book: The Semantic Tradition from Kant to Carnap
  • Online publication: 05 March 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139172240.009
Available formats
×